Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3,4,8)(2,5,6,7);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^7|a^2=e,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9176, #9178, #9304, … |
| maximal normal subgroup of | #9176, #9178, #9304, … |
| maximal quotient of | #20647, #20874, #21033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1810);